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Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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45 views

Prove that the function $F(f)=\int_a^bf(x)dx$ is continuous. [closed]

Define $d:X\times X\to \mathbb{R}$ by $$d(f,g)=\int_a^b |f(x)-g(x)|dx$$ Let $\rho$ be the usual metric on $\mathbb{R}$. Prove that the function $F:X\to\mathbb{R}$ defined $$F(f)=\int_a^bf(x)dx$$ is ...
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3answers
43 views

Example of a locally-convex topological vector space which is not metrizable

I seek an example of a locally-convex topological vector space which is not a metric space. From google I found an example LF-Space. Does there exist other examples ?
4
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35 views

Let $(A_{j})$ be a sequence of closed subsets of $X$ with $A_{j} \supseteq A_{j+1}$ for all $j \in \mathbb{N}$, then $\cap A_{j} \neq \emptyset .$

Good afternoon, I'm doing Problem III.3.5 from textbook Analysis I by Amann. Here the underlying space is metric space. Could you please verify if my proof looks fine or contains logical ...
2
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0answers
31 views

Prove that $d(i,j)$ is a metric on $\{1,..,m\}$

Let $w = \{w(i,j)\}_{1 \leq i,j \leq m}$ be an $m \times m$ symmetric matrix with non-negative real entries such that $w(i,j)=0$ if and only if $i=j$. Show that $$d(i,j)=\min\left\{\sum_{j=0}^{k-1}w(...
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26 views

Countable basis in a metric space is equivalent to being a separable metric space

Say we have a metric space space $(\Omega, d)$. Is the following true: $(\Omega, d)$ has a countable basis in a metric space $\iff$ $(\Omega, d)$ separable metric space To me the equivalence seems ...
4
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1answer
56 views

Metric on finite sets of natural numbers

I was reading this question and I had a go at proving that $d^J$ is a metric. That is, if $X,Y\subset\mathbb{N}$ are finite, define $$d^J(X,Y)=1-\frac{|X\cap Y|}{|X|+|Y|-|X\cap Y|}=1-\frac{|X\cap Y|}{|...
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2answers
67 views

In an ultrametric space, is every open set closed?

I saw the following well-known fact for ultrametric spaces Every open ball is closed. So this stimulates me to think whether this is true for open set or not. By an ultramtric space, it's a ...
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21 views

Which type of set convergence is given by $\lim_{n\to\infty}d(x,A_n)=d(x,A)$ for all $x\in X$?

Let $(X,d)$ be a metric space, let $A\subseteq X$ be a subset of $X$, and let $(A_n)_{n=1}^{\infty}$ be a sequence of subsets of $X$ such that $$ \lim_{n\to\infty}d(x,A_n)=d(x,A), \quad \forall x\in ...
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42 views

prove the set $x: d(p,x) < d(q,x)$ is open

In trying to prove this, I had R^2 in mind. But the statement to be proved is true for any metric space. I claim for any $x$ in this set, $B(x,r)$ is inside the set too for $$r= min (d(x,p), |(d(x,p) -...
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41 views

Singular matrices are Nowhere Dense

I need to show that the subset S of all singular matrices is Nowhere dense in Set of all matrices of order n. What I tried:- As I need to show closure of S has no interior point . But S is a closed ...
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2answers
34 views

Proving function is well-defined over set of equivalence classes

This is a subproblem of a problem I am trying to solve. For a measure space $(X,\mathcal{A}, \mu)$ define $\backsim$ as the relation on $\mathcal{A}$ where $A \backsim B$ iff $\mu(A \triangle B)=0$ ...
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5answers
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$d(A,B)=\min\{ d(a,b): a\in A\text{ and }b\in B\}>0$ then $A\cup B$ is not connected

If $d(A,B)=\min\{ d(a,b): a\in A\text{ and }b\in B\}>0$ then $A\cup B$ is not connected. We claim that the two sets are disjoint, suppose it is not we have $d(A,B)=0$ So now we have to say any ...
2
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1answer
34 views

Subspace of a separable metric space is also separable [duplicate]

I see a result on a textbook but I do not know how to prove it. let $(X,d)$ be a separable metric space and $A \subset X$. Suppose that the topology induced on $A$ by the metric $d_A$ coincides with ...
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29 views

Convergence of sequences in metric space

For each positive integer $m$, consider the following sequences: $x_m:=\{0,0,...0,m,m,m,...\}$ and $y_m:=\{0,...,0,\frac{1}{m}, \frac{1}{m},...\}$, where the only zero terms of these sequences are ...
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30 views

How to construct metric of direct sum of spaces?

Consider the space $\mathbb{R}^3\oplus \mathrm{SU}(2)$. How do you construct a metric for it using the metrics for the subspaces? I'm considering using something like: $d(x_1, x_2) = d_1\left(\vec{v}...
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4answers
93 views

Why is the exterior set of $\mathbb R\setminus \mathbb Q$ a null set?

Given the set $\mathbb R\setminus \mathbb Q.$ The interior set is the collection of all the interior points, where the interior point of a set $S$ from $\mathbb R,$ is a point $x \in S,$ such that ...
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18 views

Prove Q is open using the square metric.

Let $Q=\{(x,y)\in\mathbb{R}^2|y>x^2\}$. Using the square metric provide a value for $\delta$ so that for any point $(x_0,y_0)\in Q$ the ball $B_{\delta}(x_0,y_0))\subset Q$. The square metric is ...
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1answer
37 views

$\mathrm{dim}(\partial U_k)\leq n$ for all $k$, implies $\mathrm{dim}(\partial (\bigcap_k U_k))\leq n$?

Let $X$ be a locally compact, Hausdorff, second countable space. Let $(U_k)_{k\in\mathbb{N}}$ be a countable family of closed subsets in $X$ with $\mathrm{dim}(\partial_X U_k)\leq n$, for all $k$ (...
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1answer
36 views

Equality of definitions of Lower Semicontinuity [closed]

I would like to show the equality (i.e. iff) of the following definitions of lower semicontinuity without the use of liminf. Definition: Lower Semicontinuous Let $(X, ||\cdot||)$ be a normed vector ...
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1answer
57 views

How does the iron shell know about infinity?

While this question involves some physics terms, its nature is purely mathematical of differential geometry. Consider a spacetime defined by a static heavy thin shell that is somewhat larger than its ...
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4answers
33 views

Explicit value for $\delta$ in the euclidean metric.

Let $Q=\{(x,y)\in\mathbb{R}^2|y>x\}$. Using the euclidean metric give and explicit value for $\delta_(x,y)$ so that $B_\delta(x,y)$ is contained in $Q$. My work so far... Using standard calculus ...
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2answers
74 views

Consider the set $A = \{ (x, y, z) :\, z\leq 6 \}\subseteq \mathbb R^3$. Show that $A$ is closed.

Consider the set $A = \{ (x, y, z) :\, z\leq 6 \}\subseteq \mathbb R^3$. Show that $A$ is closed.
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1answer
26 views

Question about Uniform Spaec with a nested space

This question is from the book "General Topology" written by John Kelly and it is Exercise D in Chapter 6, Page 204. For definition of uniform space and the topological generated by the uniform, ...
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46 views

Reference request. Invariance of the ball by an application.

I'm writing an article and didn't want to reproduce the proof of lemma below. I just wanted to state and indicate a reference in English language to the proof of the lemma. I have a reference in ...
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1answer
34 views

What are the effects of isometries upon the periodicity of p-adic numbers?

Let $f:X\to X$ be an isometry on the 2-adic metric space $X,d$. Let the periodicity of $x$ be the number of repeating terms in its 2-adic representation. What rules govern the periodicities of $f(x)$...
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53 views

Is it possible to consistently factor powers of $3$ throughout $\Bbb Z_2$?

I'm pretty sure I was told it's impossible to identify multiples of $3$, or powers of $3$ in the p-adic ring $\Bbb Z_2$, and this claim makes total sense to me. Let $f(x)=3x+2^{\nu_2(x)}$ Is it fair ...
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7 views

If a collection of $N$ points has all Cayley-Menger determinants of order $k<N$ equal zero, are all order $m>k$ CM dets zero?

This paper on page 2284 goes through four Theorems that give necessary and sufficient conditions for a collection of points with distances given between each two points (in terms of the Euclidean ...
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0answers
25 views

Does the Cayley-Menger determinant for a tetrahedron being positive, along with the triangle inequality holding, imply Ptolemy's inequality.

I was confused after finding out, by computing counterexamples, that the Cayley-Menger determinant for a tetrahedron being positive does not actually imply that the triangle inequality and Ptolemy's ...
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1answer
40 views

Can sequences in two seperated sets have the same limit point?

A set $X$ is said to be seperated $\iff$ $\exists$ two subsets $A $ and $ B $ of $X$ such that $A \cap cl(B) \neq \emptyset $, $B \cap cl(A) \neq \emptyset$ and $A \cup B =X$. Now,we do not claim $...
2
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1answer
39 views

If $A_k \subseteq X_k$ is closed in $X_k$ for all $1 \leq k \leq m$, then $\prod_{k=1}^{m} A_k$ is closed in $\prod_{k=1}^{m} X_k$

I'm trying to prove this basic property of the product metric. Could you please verify if my proof looks fine or contains logical gaps/errors? Thank you so much for your help! If $(X_{k}, d_{k})$ are ...
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1answer
19 views

The canonical projections on metric spaces are continuous and open

Bonjour, I'm doing Problem III.2.18 from textbook Analysis I by Amann/Escher. $p$ is open if the images of open sets under $p$ are open. $X \times Y$ is endowed with the product metric. ...
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0answers
38 views

Proving that $d$ is a metric on $\mathbb{R}^X$

Let $\mathbb{R}^X$ denote the set of all functions on a non-empty set $X$ to the real line $\mathbb{R}$, and for $f,g \in \mathbb{R}^X$, let $\displaystyle d(f,g)= \sup_{x \in X } \frac{|f(x)-g(x)|}{1+...
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1answer
33 views

How to prove a function is continuous on a metric space by the theorems about continuity on a point?

I'm asking for some hints or direction for my homework of proving a function is continuous on a metric space. I was given this homework after the lecture about the continuity of functions on a point ...
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0answers
23 views

$X$= the set of all continuously differentiable functions on $[a,b]$. Show that $d(f,g)$ is a metric.

$\displaystyle d(f,g)=\sup_{a\leq x \leq b} |f(x)-g(x)|+ \sup_{a\leq x \leq b} |f'(x)-g'(x)|, \quad f,g \in X$. Approach: $|f(x)-g(x)| \leq |f(x)-h(x)|+|h(x)-g(x)| \implies$ $ |f(x)-g(x)| \leq \...
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0answers
27 views

Show that $M(3)$ of metric axioms can be replaced by a weaker axiom $M'(3)$

$M(3):=\text{ For all $x,y,z \in X$, $d(x,z) \leq d(x,y)+d(y,z)$}$ $M'(3):= \text { If $x,y,z \in X$ are distinct, then $d(x,z)\leq d(x,y)+d(y,z)$}$. [all the other axioms are identical] Proof: To ...
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0answers
41 views

Is $d(x,y)=\bigg[\sum_{n=1}^\infty(x_n-y_n)^2 \bigg]^{1/2}$ a metric on the set of all real sequences whose almost every term is zero?

The question given in my book goes exactly as follows: Let $X$ consist of all sequences $x=\{x_n\}$ of all real sequences, such that all but finitely many terms $x_n$ are zero. Prove that $d(x,y)=\...
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2answers
63 views

Metric on the cone of a compact metric space

It is known that if $X$ is a compact metric space (with bounded metric), then its cone $CX$ (the quotient $X \times [0,1] / X \times \{1\}$) is metrizable. Is there a way to provide an explicit ...
2
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1answer
27 views

Prove that $\mathbb B (x_0,r)$ is open and $\overline{\mathbb B} (x_0,r)$ is closed in $X$

Let $(X,d)$ be a metric space. For $x_0 \in X$ and $r>0$, we define the balls $\mathbb B (x_0,r)$ and $\overline{\mathbb B} (x_0,r)$ by $$\mathbb B (x_0,r) = \{x \in X \mid d(x,x_0) < r\}, \quad ...
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1answer
20 views

Problem III.2.14 in Analysis I by Amann: open (closed) functions between metric spaces

Good afternoon, I'm doing Problem III.2.14 from textbook Analysis I by Amann. My attempt: (a) Let $A \subseteq \mathbb R$ be open w.r.t $d$. For $y \in A$, the open ball $\mathbb B_\delta (y, 1/...
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0answers
46 views

How to prove or disprove:$\inf_i{\sup_x\{f_i(x)+g_i(x)}\} \leq \inf_i{\sup_x{f_i(x)}} + \inf_i{\sup_x{g_i(x)}}$

How to prove or disprove:$$\inf_i{\sup_x\{f_i(x)+g_i(x)}\} \leq \inf_i{\sup_x{f_i(x)}} + \inf_i{\sup_x{g_i(x)}}$$, because, $f_i(x) \leq \sup\limits_x{f_i(x)}$ and $g_i(x) \leq \sup\limits_x{g_i(x)}$...
4
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0answers
46 views

To prove that $\frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ is a metric on $\mathbb{C}$ [duplicate]

I am practically clueless on how to prove the triangle inequality here. The case is evident when $|z|≤|x|$ and $|z|≤|y|$. But how do I prove it in general? Any hint/answer would be much appreciated.
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0answers
16 views

Continuity of a function defined from metric space to $\mathbb{R}^k$

In my script I'm reading that if $(X,d)$ is a metric space and $Y=\mathbb{R}^k$ with norms$||\cdot ||_2, ||\cdot ||_1, ||\cdot ||_{\infty}$, then $f:X \rightarrow \mathbb{R}^k$ is continuous if and ...
4
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1answer
57 views

Union of continuous functions on closed (or open) subsets is continuous

Good morning, I'm doing Problem III.2.13 from textbook Analysis I by Amann. My attempt: Let $C \subseteq Y$ be closed in $Y$. Because $g$ is continuous, $g^{-1}[C]$ is closed in $A$. As such, ...
1
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1answer
8 views

Find $r_2$ such that $U_1(x, r_2) \subset U_2(x, \varepsilon)$ to show two metrics are equivalent.

Consider $X := (0, 1]$ and the metrics $d_1(x, y) := \left| \frac{1}{x} - \frac{1}{y} \right|$ and $d_2(x, y) = |x - y|$ and show they are topologically equivalent, i.e. for all $x \in X$ $$ \forall ...
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0answers
7 views

Does the complement of “iterated” isolated points contain isolated points?

Let $X$ be a complete separable metric space, e.g. a closed subset of $\mathbb{R}^n$. For $k=1,2,\dots$, let $I_k$ be the set of all isolated points in $X_k=X_{k-1}\setminus I_{k-1}$, with $X_0=X$, $...
1
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1answer
21 views

Strict positivity of the infimum of a set of sup-metrics

I am struggling with a subset of $\,l_\infty$ that has the following structure: Let $a>0$ and consider set of sequences $S=\{x_k:k\in\mathbb{N}\}$ such that $x_{k,k}=a$ and $x_{k,i}=0$ for all $i\...
6
votes
2answers
92 views

Word metric on a finitely generated subgroup versus the word metric of its finitely generated parent

Let $G$ be a finitely generated group and $H\subseteq G$ a finitely generated subgroup. For every finite set of a generators $F$ for $H$, we can extend to a finite generating set $\hat{F}$ for $G$. ...
1
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0answers
60 views

Is the set of positive functions open in $C_b(\mathbb{R})$?

Let $C_b(\mathbb{R})$ stand for the set of all continuous and bounded self-maps on $\mathbb{R}$, and view this as a metric subspace of $B(\mathbb{R})$, the set of all functions on real numbers. Is $\{...
2
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0answers
28 views

If $X$ is complete, then $A$ is complete if and only if $A$ is closed in $X$

I'm doing Problem III.2.21 from textbook Analysis I by Amann. Could you please verify if my proof look fine or contains logical gaps/errors? Any suggestion is greatly appreciated. Thank you so ...
2
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1answer
52 views

Show that the function $f : A \cup B \to Y, \quad x \mapsto \begin{cases} g(x), &x\in A \\h(x), &x\in B\end{cases}$ is continuous.

Good evening, I'm doing Problem III.2.13 from textbook Analysis I by Amann. Could you please verify if my proof look fine or contains logical gaps/errors? Any suggestion is greatly appreciated. ...